Best way to make cartoon style hair?
There are a number of video tutorials on this subject that will provide a much better answer than we can here. I would suggest this one by Kent Trammel from CG Cookie:

https://www.youtube.com/watch?v=Nn7mISvVzJk

The result isn't exactly like the image you posted, but is similar enough that you should be able to adapt the techniques to your situation.

The other way to make hair like you want is to sculpt it. There are a couple of good tutorials online about how to do that. Even tutorials for other programs like ZBrush can be a good way to find out how to do it. CG Cookie has a great tutorial on how to sculpt hair, but it appears that a subscription of sorts is required for this one (might be free, not sure). Here is the trailer for the video. 

https://www.youtube.com/watch?v=gXKYN5qJX1c